A series of real numbers is said to be conditionally convergent if it is convergent but not absolutely convergent.
By rearranging the terms of a conditionally convergent series we can make the rearranged series converge to any real number (or diverge to $\pm\infty$). This is known as the Riemann series theorem. I think it is a nice theorem, but since I had never used it or seen it used (great is my ignorance!) it was in my box of results that are out of my mind. It had been there for years when, out of the blue, I used it as a possible way to solve this problem.
Regardless of whether or not I used the theorem correctly, this unforseen re-encounter with a dusty result I had almost forgotten made me wonder about the beauty of mathematics and how it never ceases to surprise us. I have to reconsider that silly box I mentioned above!
I would like to know about uses of the Riemann series theorem.
I would like to know of similar experiences. A result that after time fell into your box of unrecalled things only to be re-encountered later with joy.
Ok it's a bit late - 8 years!, but there is a use of the Riemann series theorem. In Spivak's Calculus on manifolds, there is a theorem about partitions of unity. An important aspect of partitions of unity is defining an extended integral using them. It is important that the definition of this partition of unity is consistent - that the integral doesn't change when we use a different partition of unity. For this to hold, we need the extended definition of the integral to be absolutely summable. If this does not hold, and the sequence is conditionally convergent, then the extended integral can give different values depending on the partition of unity given. This is the essence of exercise 3-38 in Calculus on Manifolds